Perceiving Geometry: Geometrical Illusions Explained by Natural Scene Statistics PDF
Preview Perceiving Geometry: Geometrical Illusions Explained by Natural Scene Statistics
Perceiving Geometry Geometrical Illusions Explained by Natural Scene Statistics Perceiving Geometry Geometrical Illusions Explained by Natural Scene Statistics by Catherine Q. Howe and Dale Purves LibraryofCongressCataloging-in-PublicationData PerceivingGeometry:GeometricalIllusionsExplainedbyNaturalSceneStatistics/ byCatherineQ.Howe,DalePurves AC.I.P.Cataloguerecordforthisbookisavailable fromtheLibraryofCongress. ISBN-10:0-387-25487-0 (alkpaper) e-ISBN-10:0-387-25488-9 ISBN-13:978-0387-25487-6 e-ISBN-13:978-0387-25488-3 (cid:1)C 2005SpringerScience+BusinessMedia,Inc. Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithout thewrittenpermissionofthepublisher(SpringerScience+BusinessMedia,Inc.,233Spring Street,NewYork,NY10013,USA),exceptforbriefexcerptsinconnectionwithreviews orscholarlyanalysis.Useinconnectionwithanyformofinformationstorageandretrieval, electronicadaptation,computersoftware,orbysimilarordissimilarmethodologynowknow orhereafterdevelopedisforbidden. Theuseinthispublicationoftradenames,trademarks,servicemarksandsimilarterms,even iftheyarenotidentifiedassuch,isnottobetakenasanexpressionofopinionastowhether ornottheyaresubjecttoproprietaryrights. PrintedintheUnitedStatesofAmerica. 9 8 7 6 5 4 3 2 1 SPIN11385059 springeronline.com Contents Acknowledgment...................................................... vii 1. Introduction ....................................................... 01 2. TheGeometryofNaturalScenes..................................... 15 3. LineLength ....................................................... 25 4. Angles ............................................................ 37 5. Size............................................................... 47 6. Distance .......................................................... 63 7. TheMu¨ller-LyerIllusion............................................ 73 8. ThePoggendorffIllusion............................................ 85 9. Implications ....................................................... 97 References............................................................ 107 Glossary.............................................................. 115 Index................................................................. 125 Acknowledgment WearegratefultoourcolleaguesShuroNundyandZhiyongYangfortheircollaboration insomeoftheworkdiscussedinthebook,andformuchusefuladviceanddiscussion. HelpfulcriticismandadvicewasalsoprovidedbyJimVoyvodic. Chapter 1 Introduction Understandingvision—whetherfromaneurobiological,psychologicalorphilosophi- calperspective—representsadauntingchallengethathasbeenpursuedformillennia. Duringatleastthelastfewcenturies,naturalphilosophers,andmorerecentlyvisionsci- entists,haverecognizedthatafundamentalprobleminbiologicalvision—andindeed afundamentalprobleminperceptiongenerally—isthatthesourcesunderlyingvisual stimuliareunknowableinanydirectsense. Thereasonforthisquandaryistheinherentambiguityofthestimulithatimpinge onsensoryreceptors.Inthecaseofvision,thelightthatreachestheeyefromanyscene conflatesthecontributionsofreflectance,illuminationandtransmittance,aswellasa hostofsubsidiaryfactorsthataffecttheseprimaryphysicalparameters(Figure1.1A). Evenmoreimportantwithrespecttothetopicunderconsiderationhere,spatialprop- ertiessuchasthesize,distanceandorientationofphysicalobjectsarealsoconflated in light stimuli (Figure 1.1B). As a result, the provenance of light reaching the eye at any moment—and therefore the significance of the stimulus for visually guided behavior—isprofoundlyuncertain.Inmoreformalterms,thisquandaryisreferredto astheinverseopticsproblem. Thesebasicfactsabouttherelationshipoftherealworldandtheinformationcon- veyedbylightreachingtheretinapresentadifficultproblem.Successfulbehaviorina complexandpotentiallyhostileenvironmentclearlydependsonrespondingappropri- atelytothephysicalsourcesofvisualstimuliratherthantothephysicalcharacteristics ofstimuliassuch(whichasindicatedinFigure1.1,areofuncertainsignificance).If theretinalimagesgeneratedbylightstimulicannotspecifytheunderlyingrealityan observer must deal with, how then does the visual system produce behavior that is generallysuccessful? EXPLORINGVISIONINTERMSOFPERCEIVEDGEOMETRY In the chapters that follow, we consider the evidence that, with respect to space, the humanvisualsystemsolvesthisproblembyincorporatingpasthumanexperienceof 2 PerceivingGeometry:GeometricalIllusionsExplainedbyNaturalSceneStatistics A SSttiimmuulluuss Stimulus Illumination Reflectance Transmittance B Retinal projection Figure1.1 Thefundamentalprobleminbiologicalvisionisthenecessarilyuncertainrelationshipbetween theinformationintheimagesthatfallontheretinaandtheirreal-worldsources.A)Conflationofthefactors thatdeterminetheamountandspectralqualityoflightfallingontheretina.Illuminationdependsonthe propertiesofasourcelikethesun;thereflectanceofobjectsdependsontheirphysicalcomposition;and transmittancedependsontheamountandqualityoftheatmosphereinterveningbetweenanobjectandthe observer(aswellasbetweenthesourceofilluminationandtheilluminatedobjects).Thesebasicfactorsthat togetherdeterminetheluminanceandspectraldistributionofanystimulusattheeyecannotbedisentangled byanalysisoftheretinalimage.B)Theproblemismuchthesameintheperceptionofgeometry,sincethe spatialpropertiesofthree-dimensionalobjectsarealsoconflatedwhenlightarisingfromthemisprojected ontoaplane.Thediagramshowsthatthesameretinalprojectioncanbegeneratedbyobjectsofdifferent sizesatdifferentdistancesfromtheobserver,andindifferentorientations.Again,thereisnologicalwayto disentanglethesefactorsbyanalysisoftheretinalimage.(AfterPurvesandLotto,2003) Introduction 3 with what projected patterns of light on the retina have typically corresponded to in the real world. Indeed, we consider this not simply an adjunct to vision, but the fundamental scheme that determines spatial vision. This empirical strategy explains manyotherwisepuzzlingaspectsofwhatwesee,andtheseexplanations—whetherin qualitative or quantitative terms—provide the best indication to date of how human perceptionsofthegeometricalaspectsoftheworldareactuallygenerated. Thecruxoftheargumentisthatthelinkbetweenstimuliandperceptswithrespect tovisualspace—i.e.,thewayweexperiencesize,distanceandorientation—canonly be understood in a statistical framework in which the perceptions generated by light patternsprojectedontotheretinaaredeterminedbytheprobabilitydistributionsofthe possiblesourcesofthoseprojections.Thisframeworkcanrationalizemanyotherwise puzzlingdiscrepanciesbetweenvisualperceptsandthephysicalparametersofvisual stimuli. These discrepancies—often presented in the form of “geometrical illusions” (seeFigure1.2)—havelongpresentedachallengetoanyoneinterestedinthenatureof vision,andattemptstounderstandthemcanbetracedbackseveralcenturiesormore. Giventheinherentambiguityofretinalimages,thebiologicalrationaleforseeing the geometrical aspects of retinal stimuli in terms of the probability distributions of their possible sources is not difficult to understand. Much to the advantage of the observer,thisvisualstrategycontendswiththeproblemofstimulusambiguitybytaking A B C D E Figure1.2 Examplesofsomemuch-studiedgeometricalillusions.A)TheT-illusion.B)TheHering illusion.C)ThePonzoillusion.D)TheMu¨ller-Lyerillusion.E)ThePoggendorffillusion. 4 PerceivingGeometry:GeometricalIllusionsExplainedbyNaturalSceneStatistics advantageofeonsoftrialanderrorinhumanexperience.Inthiswaywe,andpresumably allotheranimalswithsophisticatedvision,ensurethatvisuallyguidedresponseswill usuallydealsuccessfullywithobjectsandconditionsthatareunknowablebyanydirect means. A consequence of generating percepts on a statistical basis, however, is that whatobserversactuallyseedoesnotalwayscorrespondtothephysicalcharacteristics ofthestimulusortheparticularphysicalconditionsthatgeneratedthestimulus;rather, whatisseencorrespondstotheempiricalsignificanceofthestimulus,i.e.,whatithas typicallymeantforvisuallyguidedbehavior. Hereweexaminethevalidityofthisgeneralidea,usingasexamplesaseriesof classicalgeometricalstimuliandthe“illusions”theygenerate(thequotationmarksare tosuggestthatthisisnotaparticularlyaptwordsince,inthepresentframework,all visualperceptsareequallyconstructedfromthestatisticalinformationacquiredthrough experience). The approach in each instance is to examine the statistical relationship between the relevant retinal images and their real-world sources, asking whether the perceptsreportedbyhumansubjectsaccordwiththepredictionsmadeonthebasisof the statistics derived from a database of natural scenes. The database includes mea- surements of the distance and direction of the physical sources of each point in the images,andiseffectivelyaproxyfortheaccumulatedvisualexperienceofourspecies. Assuch,thedatabase(seeChapter2)canbeusedtorevealthestatisticalregularities betweenretinalimagesandreal-worldsourcesthatmusthavedeterminedtheevolution ofhumanvision. Of course, a variety of other visual perceptual qualities can be (and have been) exploredinthisframework,includingbrightness/lightness,colorandsomeaspectsof motion(reviewedinPurvesandLotto,2003).Wehavechosentofocushereonperceived spatial relationships because understanding the way we see geometry is intrinsically interestingandmuchdebated;afurtherreasonisthat,fromatechnicalperspective,the probabilitydistributionsofthepossiblesourcesofgeometricalstimulicanbederived directly by an analysis of range images of natural scenes. The ability to determine the statistical relationship between geometrical projections and their sources is an enormousadvantage,andmeetingthisgoalismuchmoredifficult(althoughcertainly notimpossible)withrespecttootherstimuluscharacteristics. Of course, the fact that geometrical stimuli and the perceptual anomalies they elicit have challenged the imaginations of so many thinkers in a variety disciplines oversomanyyearsisstrongmotivationaswell.Intheend,however,thepointofthe worksummarizedhereissimplytounderstandhowandwhyweperceivevisualspace andthegeometryoftheobjectsthereinthewaywedo. EXAMPLESOFWELL-KNOWNGEOMETRICALILLUSIONS Numerousobservershavepointedoutthatmeasurementsmadewithrulersorprotrac- tors of a variety of simple visual stimuli are often at odds with the perceptions they elicit,frequentlyinamostengagingway.Constructingstimulithatproduceapartic- ularlyintriguinggeometricalillusionwassomethingofacottageindustryinthe19th and early 20th centuries; although some of these demonstrations were described by thepreeminentvisionscientistsofthetime,agoodgeometricalillusionhasprovided
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